lim(x→+∞)[tan √(2004 + x) – tan √x] = ?

lim(x→+∞)[tan √(2004 + x) – tan √x]　does not converge to any number.

Let function abs(x) be absolute value of x ignoring the sign,
e.g. abs(-1)=1, abs(2)=2

Assume the limit converge to some number, let it be l,
For some small number e, let say 0.001,
there exist a big number N such that for all n>N
abs(tan √(2004 + n) – tan √n - l) <= e

This could not be possible because
a) we can setup m>n such that √m is just slightly less than (k+0.5) pi for some integer k, and so that tan √(2004 + m) – tan √m is a big negative number.
b) we can also setup o>n such that √(2004 + o) is just slightly less than (k+0.5) pi for some integer k, and so that tan √(2004 + o) – tan √o is a big positive number.

That means the value of (tan √(2004 + n) – tan √n) has to be various from very big negative to very big positive, repeatingly, and cannot coverge for the range n>N.

2007-03-07 23:20:01 補充：
回上面答者：因為 cos√xcos√(2004 + x) 可以係零，也可以係極小正或極小負數，這令上述答案有缺陷與不確定性。必需證明 cos√xcos√(2004 + x) 無法接近零，我才能同意第一個回答。